From fbbbf6fa6944b1b34b8b47a7f7e0848b4dc934db Mon Sep 17 00:00:00 2001 From: Grant Sanderson Date: Thu, 25 Feb 2021 08:53:12 -0800 Subject: [PATCH] Beginning matrix_exp project --- _2021/matrix_exp.py | 297 +++++++++++++++++++++++++++++++ custom/characters/pi_creature.py | 8 +- custom_config.yml | 14 +- program.prof | Bin 1308959 -> 0 bytes 4 files changed, 308 insertions(+), 11 deletions(-) create mode 100644 _2021/matrix_exp.py delete mode 100644 program.prof diff --git a/_2021/matrix_exp.py b/_2021/matrix_exp.py new file mode 100644 index 0000000..ed56089 --- /dev/null +++ b/_2021/matrix_exp.py @@ -0,0 +1,297 @@ +from manim_imports_ext import * + + +def get_integer_matrix_exponential(matrix): + pass + + +def get_vector_field_and_stream_lines(func, coordinate_system, + magnitude_range=(0.5, 4), + vector_opacity=0.75, + vector_thickness=0.03, + color_by_magnitude=True, + line_color=GREY_A, + line_width=3, + sample_freq=3, + n_samples_per_line=8, + arc_len=3, + time_width=0.5, + ): + vector_field = VectorField( + func, coordinate_system, + magnitude_range=magnitude_range, + vector_config={ + "fill_opacity": vector_opacity, + "thickness": vector_thickness, + } + ) + stream_lines = StreamLines( + func, coordinate_system, + step_multiple=1.0 / sample_freq, + n_samples_per_line=n_samples_per_line, + arc_len=arc_len, + magnitude_range=magnitude_range, + color_by_magnitude=color_by_magnitude, + stroke_color=line_color, + stroke_width=line_width, + ) + animated_lines = AnimatedStreamLines( + stream_lines, + line_anim_config={ + "time_width": time_width, + }, + ) + + return vector_field, animated_lines + + +# Scenes + +class ShowConfusionAtMatrixExponenent(Scene): + def construct(self): + # Sticking a matrix in an exponent like this might strike you as... + base = Tex("e") + base.set_height(1.0) + matrix = IntegerMatrix( + [[3, 1, 4], + [1, 5, 9], + [2, 6, 5]], + ) + matrix.move_to(base.get_corner(UR), DL) + matrix_exp = VGroup(base, matrix) + matrix_exp.set_height(2) + matrix_exp.to_corner(UL) + matrix_exp.shift(3 * RIGHT) + + randy = Randolph() + randy.set_height(2) + randy.to_corner(DL) + + matrix.save_state() + matrix.center() + matrix.set_height(2.5) + + self.add(randy) + self.play( + randy.animate.change("pondering", matrix), + Write(matrix.get_brackets()), + ShowIncreasingSubsets(matrix.get_entries()), + ) + self.play(Blink(randy)) + self.play( + matrix.animate.restore(), + Write(base), + randy.animate.change("erm", base), + ) + self.play(Blink(randy)) + + # ...odd, to say the least. + rhs = Tex("= e \\cdot e \\dots e \\cdot e") + rhs.set_height(0.75 * base.get_height()) + rhs.next_to(matrix_exp, RIGHT) + rhs.align_to(base, DOWN) + brace = Brace(rhs[0][1:], DOWN) + matrix_copy = matrix.copy() + matrix_copy.scale(0.5) + brace_label = VGroup( + matrix.copy().scale(0.5), + Text("times?") + ) + brace_label.arrange(RIGHT) + brace_label.next_to(brace, DOWN, SMALL_BUFF) + + bubble = randy.get_bubble( + TexText("I'm sorry,\\\\what?!").scale(0.75), + height=2, + width=3, + bubble_class=SpeechBubble, + ) + + self.play( + TransformMatchingParts( + base.copy(), rhs, + path_arc=10 * DEGREES, + lag_ratio=0.01, + ), + GrowFromCenter(brace), + ReplacementTransform( + matrix.copy(), brace_label[0], + path_arc=30 * DEGREES, + run_time=2, + rate_func=squish_rate_func(smooth, 0.3, 1), + ), + Write( + brace_label[1], + run_time=2, + rate_func=squish_rate_func(smooth, 0.5, 1), + ), + ) + self.play( + randy.animate.change("angry", rhs), + ShowCreation(bubble), + Write(bubble.content), + ) + self.wait() + + false_equation = VGroup( + matrix_exp, rhs, brace, brace_label + ) + + # The short version of this video would be to say the notation is misleading, + # having little to do with the number e being multiplying it by itself. + morty = Mortimer() + morty.match_height(randy) + morty.to_corner(DR) + false_equation.generate_target() + false_equation.target.scale(0.5) + false_equation.target.next_to(morty, UL) + fe_rect = SurroundingRectangle(false_equation.target) + fe_rect.set_color(GREY_BROWN) + cross = Cross(false_equation.target[1]) + cross.set_stroke(RED, 5) + nonsense = Text("This would be nonsense") + nonsense.match_width(fe_rect) + nonsense.next_to(fe_rect, UP) + nonsense.set_color(RED) + + randy.bubble = bubble + self.play( + MoveToTarget(false_equation), + RemovePiCreatureBubble(randy, target_mode="hesitant"), + morty.animate.change("raise_right_hand"), + ShowCreation(fe_rect), + VFadeIn(morty), + ) + self.play( + ShowCreation(cross), + FadeIn(nonsense), + ) + self.play(Blink(morty)) + self.wait() + + # Writing e with a matrix up top is shorthand for plugging in the matrix to a certain infinite polynomial. + real_equation = Tex( + "e^X := X^0 + X^1 + \\frac{1}{2} X^2 + \\frac{1}{6} X^3 + \\cdots + \\frac{1}{n!} X^n + \\cdots", + isolate=["X"] + ) + xs = real_equation.get_parts_by_tex("X") + xs.set_color(TEAL) + real_equation.set_width(FRAME_WIDTH - 2.0) + real_equation.to_edge(UP) + + by_definition = Text("(by definition)", font_size=24) + by_definition.set_fill(GREY_B) + bd_arrow = Vector(0.5 * UP) + bd_arrow.match_color(by_definition) + bd_arrow.next_to(real_equation.get_part_by_tex("="), DOWN, SMALL_BUFF) + by_definition.next_to(bd_arrow, DOWN) + + self.play( + TransformFromCopy(base, real_equation[0]), + FadeTransform(matrix.copy(), real_equation[1]), + ) + self.play( + GrowArrow(bd_arrow), + FadeIn(by_definition, DOWN), + Write(real_equation[2]), + FadeTransformPieces(xs[:1].copy(), xs[1:], path_arc=20 * DEGREES), + LaggedStart(*( + FadeIn(part) + for part in real_equation[4:] + if part not in xs + )) + ) + self.add(real_equation) + self.play( + randy.animate.change("pondering", real_equation), + morty.animate.change("pondering", real_equation), + ) + self.wait() + + # Still, that’s a complicated thing to do, which takes some explaining in its own right, + # and without context this definition does very little to explain how to think about this operation. + rhs_tex = "X^0 + X^1 + \\frac{1}{2} X^2 + \\frac{1}{6} X^3 + \\cdots + \\frac{1}{n!} X^n + \\cdots" + mat_tex = "\\left[ \\begin{array}{ccc} 3 & 1 & 4 \\\\ 1 & 5 & 9 \\\\ 2 & 6 & 5 \\end{array} \\right]" + ex_rhs = Tex( + rhs_tex.replace("X", mat_tex), + tex_to_color_map={mat_tex: TEAL}, + ) + ex_rhs.scale(0.5) + + ex_eq = VGroup(matrix_exp.copy(), Tex("="), ex_rhs) + ex_eq[0][1].set_color(TEAL) + ex_eq.arrange(RIGHT) + ex_rhs.align_to(ex_eq[0], DOWN) + ex_eq = VGroup(ex_eq[0], ex_eq[1], *ex_rhs) + ex_eq[1:].shift(0.1 * DOWN) + ex_eq.next_to(real_equation, DOWN, buff=2, aligned_edge=DOWN) + + false_group = VGroup(false_equation, fe_rect, cross, nonsense) + complicated = Text("Rather complicated!") + complicated.set_color(YELLOW) + complicated.next_to(ex_eq, DOWN, LARGE_BUFF) + + self.play( + TransformFromCopy(matrix_exp, ex_eq[0]), + FadeOut(false_group, DOWN), + FadeTransformPieces( + real_equation.copy()[2:], ex_eq[1:], run_time=2, + ), + randy.animate.change("hesitant"), + morty.animate.change("raise_right_hand"), + ) + self.play(Blink(randy)) + self.play( + FadeIn(complicated, shift=DOWN), + randy.animate.change("confused"), + morty.animate.change("hesitant"), + ) + self.wait() + + +class CircularPhaseFlow(Scene): + def construct(self): + plane = NumberPlane( + x_range=[-4, 4], + y_range=[-2, 2], + height=8, + width=16, + ) + plane.add_coordinate_labels() + + vector_field, animated_lines = get_vector_field_and_stream_lines( + self.func, plane + ) + + self.add(plane) + self.add(vector_field) + self.add(animated_lines) + + self.wait(10) + + # + self.embed() + + def func(self, x, y): + return (-y, x) + + def get_label(self): + pass + + +class HyperbolicPhaseFlow(CircularPhaseFlow): + def func(self, x, y): + return (x, -y) + + +class WhyWedCare(Scene): + def construct(self): + pass + # ...how to think about this operation, or more importantly why we’d care. + + # So if it’s alright with you I think we should hold off explaining this definition, or justifying the + # abuse of notation, until it’s a bit clearer what problems it helps us to solve. + + + # + self.embed() diff --git a/custom/characters/pi_creature.py b/custom/characters/pi_creature.py index f919e04..61f4275 100644 --- a/custom/characters/pi_creature.py +++ b/custom/characters/pi_creature.py @@ -8,7 +8,7 @@ from manimlib.mobject.mobject import Mobject from manimlib.mobject.geometry import Circle from manimlib.mobject.svg.drawings import ThoughtBubble from manimlib.mobject.svg.svg_mobject import SVGMobject -from manimlib.mobject.svg.tex_mobject import TexText +from manimlib.mobject.svg.text_mobject import Text from manimlib.mobject.types.vectorized_mobject import VGroup from manimlib.mobject.types.vectorized_mobject import VMobject from manimlib.utils.config_ops import digest_config @@ -224,14 +224,14 @@ class PiCreature(SVGMobject): self.to_corner(DOWN + LEFT, **kwargs) return self - def get_bubble(self, *content, **kwargs): + def get_bubble(self, content, **kwargs): bubble_class = kwargs.get("bubble_class", ThoughtBubble) bubble = bubble_class(**kwargs) if len(content) > 0: if isinstance(content[0], str): - content_mob = TexText(*content) + content_mob = Text(content) else: - content_mob = content[0] + content_mob = content bubble.add_content(content_mob) if "height" not in kwargs and "width" not in kwargs: bubble.resize_to_content() diff --git a/custom_config.yml b/custom_config.yml index c3155fb..7c2d8c7 100644 --- a/custom_config.yml +++ b/custom_config.yml @@ -1,11 +1,11 @@ directories: mirror_module_path: True - output: "/Users/grant/Dropbox/3Blue1Brown Team Folder/videos" - raster_images: "/Users/grant/Dropbox/3Blue1Brown Team Folder/images/raster" - vector_images: "/Users/grant/Dropbox/3Blue1Brown Team Folder/images/vector" - pi_creature_images: "/Users/grant/Dropbox/3Blue1Brown Team Folder/images/pi_creature" - sounds: "/Users/grant/Dropbox/3Blue1Brown Team Folder/sounds" - data: "/Users/grant/Dropbox/3Blue1Brown Team Folder/data" + output: "/Users/grant/Dropbox/3Blue1Brown/videos" + raster_images: "/Users/grant/Dropbox/3Blue1Brown/images/raster" + vector_images: "/Users/grant/Dropbox/3Blue1Brown/images/vector" + pi_creature_images: "/Users/grant/Dropbox/3Blue1Brown/images/pi_creature" + sounds: "/Users/grant/Dropbox/3Blue1Brown/sounds" + data: "/Users/grant/Dropbox/3Blue1Brown/data" temporary_storage: "" universal_import_line: "from manim_imports_ext import *" # tex: @@ -17,5 +17,5 @@ style: background_color: "#000000" camera_qualities: default_quality: "ultra_high" -window_position: UL +window_position: UR window_monitor: 1 \ No newline at end of file diff --git a/program.prof b/program.prof deleted file mode 100644 index d86df21bd512d792620ee86dd4bf9b0fca011ec2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1308959 zcmcG12Ygh;_CH7qgx&-MfgmMP7AXRPyJAp4L69b>xNMTm&4pw)+}!{Xe8z%}_po4Z zs1Lg$Dquqt3)l-5aP4AS3qBhviu}LdnLBsx&0R3R)&Ki^-gwB)+?hFZ=FFMX=J))1 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